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How many solutions does the system of equations below have?
{:[x+6y+8=0],[2x+11=-12 y]:}
(A) no solution
(B) one solution
(C) infinitely many solutions

How many solutions does the system of equations below have? $\newline$ $\begin{array}{l} x+6 y+8=0 \\ 2 x+11=-12 y \end{array}$ $\newline$ (A) no solution $\newline$ (B) one solution $\newline$ (C) infinitely many solutions

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Find the number of solutions to a system of equations

Full solution

Q. How many solutions does the system of equations below have?

\newline

\begin{array}{l} x+6 y+8=0 \\ 2 x+11=-12 y \end{array}

\newline

(A) no solution

\newline

(B) one solution

\newline

(C) infinitely many solutions

Write Equations: Write down the system of equations: $\newline$ $x + 6y + 8 = 0$ $\newline$ $2x + 11 = -12y$ $\newline$ Are the equations in the same form? $\newline$ No, the second equation needs to be rearranged to match the form of the first equation.
Rearrange Second Equation: Rearrange the second equation to match the form of the first equation: $\newline$ $2x + 11 = -12y$ $\newline$ Move all terms involving variables to the left side and constants to the right side: $\newline$ $2x + 12y = -11$ $\newline$ Now the system of equations is: $\newline$ $x + 6y + 8 = 0$ $\newline$ $2x + 12y = -11$
Compare Coefficients: Compare the coefficients of the two equations: $\newline$ The first equation has coefficients $1$ for $x$ and $6$ for $y$ . $\newline$ The second equation has coefficients $2$ for $x$ and $12$ for $y$ . $\newline$ Are the ratios of the coefficients of $x$ and $y$ the same in both equations? $\newline$ Yes, the ratio is $x$ $0$ in the first equation and $x$ $1$ in the second equation, which simplifies to $x$ $0$ .
Compare Constants: Since the ratios of the coefficients are the same, the lines are parallel or the same line. To determine which, compare the constants on the right side of the equations: $\newline$ The first equation has a constant of $-8$ . $\newline$ The second equation has a constant of $-11$ . $\newline$ Are the constants proportional to the coefficients of $x$ and $y$ ? $\newline$ No, the constants are not proportional to the coefficients of $x$ and $y$ .
Determine Number of Solutions: Determine the number of solutions to the system of equations: $\newline$ Since the lines have the same ratios of coefficients but different constants, they are parallel and do not intersect. $\newline$ Therefore, the system of equations has no solution.

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